Answer:
Explanation:
To find the domain of the equation f(x) = (x^2+4x-3)/(x^4-5x^2+4), we need to determine the values of x for which the equation is defined.
The domain of a function is the set of all possible input values (x-values) for which the function is defined. In this case, we have a rational function, which means we need to consider two things:
1. The denominator cannot equal zero, as division by zero is undefined.
2. The square root of a negative number is not a real number, so the radicand (expression inside the square root) cannot be negative.
To find the values of x that make the denominator equal to zero, we set the denominator equal to zero and solve for x:
x^4 - 5x^2 + 4 = 0
We can factor this equation as follows:
(x^2 - 1)(x^2 - 4) = 0
Using the difference of squares, we further factor:
(x - 1)(x + 1)(x - 2)(x + 2) = 0
Setting each factor equal to zero, we get:
x - 1 = 0, x + 1 = 0, x - 2 = 0, x + 2 = 0
Solving these equations, we find:
x = 1, x = -1, x = 2, x = -2
These are the values of x that make the denominator equal to zero. Therefore, they are not included in the domain of the function.
The domain of the function f(x) = (x^2+4x-3)/(x^4-5x^2+4) is all real numbers except x = 1, x = -1, x = 2, and x = -2.
In interval notation, the domain can be expressed as (-∞, -2) ∪ (-2, -1) ∪ (-1, 1) ∪ (1, 2) ∪ (2, ∞